# Tagged Questions

This tag is for questions about the foundations of mathematics, and the formalization of mathematical concepts in foundational theories (e.g. set theory, category theory, and type theory).

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### Can these definitions of the words “problem” and “solution” be formalized, and if so, has this been done? If so, where can I learn more about it?

I had a thought. Define that: Vague Definition 0. A problem consists of: a set $X$ a set $Y$ a function $f : X \rightarrow Y$ a way $\overline{X}$ of representing the elements of $X$ ...
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### How can mathematics work in wildly different set theories?

There are several set theories, e.g. ZFC and NF, which often have different axioms or are even outright contradictory. And yet most of other branches of mathematics, e.g. abstract algebra or topology, ...
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### Is PA the most common foundation for arithmetic?

Are Peano Axioms the most common and widely accepted axiomatization of arithmetic, just as ZFC is the most common and widely accepted foundation for all of mathematics?
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### What type of reasoning is employed when studying metalogic?

When studying, or doing any kind of reasoning about logic, what type of logic is used? Or how is the reasoning structured?
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### How are image and pre-image different from range and domain respectively?

How are image and pre-image different from range and domain respectively, in Layman's terms (as simple as possible)? Are they basically just keywords that often indicate more nuanced subsets of the ...
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### Is this why proving Con(ZFC) is impossible? [duplicate]

I've stumbled upon this post on MathOverflow, and the poster has something interesting to say. In short: if we were to come up with a mathematical proof of the consistency of ZFC, we would be able ...
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### Is PA the first axiomatization of arithmetic to be discovered? [closed]

Is Peano Arithmetic the first axiomatization of arithmetic to be discovered?
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### Why is it impossible to prove absolute consistency of a theory falling prey to Godel's theorems?

Why is it impossible to prove absolute consistency of a theory T falling prey to Godel's theorems? I understand that a theory falling prey to Godel's second incompleteness theorem cannot prove its own ...
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### Need an assistance with a specific step of a specific Division Algorithm proof

I'm trying to wrap my head around a Division Algorithm's proof. That is, Let $a, b \in \mathbb{Z}, a \neq 0$. Then there are unique $q,r \in \mathbb{Z}$ such that $b = qa + r, 0 \leq r < |a|$. ...
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### Why doesn't this definition of natural numbers hold up in axiomatic set theory?

I was reading about older definitions of the natural numbers on Wikipedia here (in retrospect, not the best place to learn mathematics) and came across the following definition for the natural ...
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### Impossibility of proving a foundation to be consistent

An argument came to my mind that seems really mind-blowing and I haven't found it anywhere. Here's how it goes: We call a formal system F embodied in classical logic a foundation of mathematics when ...
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### Divide by 0 alternative [closed]

Cutting to the chase. I know you can't divide by zero. And I have read a good few explications for this. And I am happy with this as a fact. BUT my question is based on this: X / N = A "should ...
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### Logic without content? Takeuti and Zaring

I am reading the book "Introduction to Axiomatic Set Theory" by Takeuti and Zaring, and I wonder if I understand the "language of logic" from chapter 2 properly. They write: The language of our ...
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To found the current set theory, it has been necessary to remove some paradoxes like the well-known Russel paradox. It was thus necessary to clarify why things like $$\{x : x \notin x\}$$ can't ...
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### Constructing natural numbers as lists of units (possible infinite objects)

I'm puzzled by this question, which is more about relation between two type theoretic approaches. Nevertheless, It can be shortened to the question : When it is correct (if ever) to construct ...
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### Axioms of Trigonometry

On Wikipedia it gives a picture of all trigonometric functions of an angle laid atop the unit circle, 1. Obviously there are other trigonometric identities, but what I'm wondering is, does ...
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### Godel's proof for dummies [closed]

Can someone give me as simple-a-proof as possible for Godel's Incompleteness Theorem? I'd love to understand it more.
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### Why might Dieudonne have been “begging the question” by appealing to second-order Peano Axioms?

Following a comment by Peter Smith, I've been reading A. R. D. Mathias's paper The Ignorance of Bourbaki. Parts of the paper are above my head, but I understand it well enough for my own amateurish ...
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### Classes of binary operations between functions

Let $f,g : D\to \mathbb{R}$ be two functions defined from a domain $D\in \mathbb{R}$ to $\mathbb{R}$. I am looking for classes of binary operations $\circ$ between $f$ and $g$ that produce an ...
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### How can I prove that if $a^7 = b^7$ then $a=b$, with $a,b \in \mathbb{Z}$

I've tried with divisibility, meaning that since $a$ divides $a^7$, then $a$ divides $b^7$ and in the same way b divides $a^7$, but I can't seem to go further than this. What properties of the ...
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### Understanding foundational terms: notions, objects and meta-objects

I am trying to take my problem solving skill to next level. It looks like It takes a lot of mathematical discipline. Here, This post buys me to get better at proof writing. So, I think is useful to ...
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### Axioms of motion (Redei version)

I try to understand the axioms of motion in Redei's "Foundationd of Euclidean and Non-Euclidean Geometries, according to F. Klein" 1968 Redei gives as Axioms: Any motion is a one to one mapping of ...
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### Are the assertions “$2 + 2$ equals $4$” and “$2 +2$ is $4$” identical

Are the assertions "$2 + 2$ equals $4$" and "$2 +2$ is $4$" identical? Or is this a linguistic, psychological or murky philosophical thing rather than a mathematical thing
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### Foundations of Differential Calculus

In the preface of Foundations of Differential Calculus there's a section that says: Thus, if the quantity $x$ is given an increment $\omega$, so that it becomes $x + \omega$, its square $x^2$ ...
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### What does “Mathematics of Computation” mean?

I visited this link: http://www.ams.org/journals/mcom/1950-04-030/S0025-5718-50-99474-9/ And I a bit confused by its title "Mathematics of Computation". I am not a native English speaker. Could ...
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### Will Homotopy Type Theory ever be as accessible as traditional Set Theory?

At the moment, Homotopy Type Theory is barely accessible to undergraduates, and only the most advanced or most gifted could have a decent chance of grasping it at a workable level without mountains of ...
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### Yoneda Lemma and foundations

Yoneda Lemma says that, for a locally small category $\mathcal{C}$, an object $A$ in $\mathcal{C}$ and a functor $F:\mathcal{C}^{op}\to \textbf{Set}$, the natural transformations $[Hom(-,A):F]$ is in ...
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### Bijective map on $(\Bbb N \times\Bbb N)/R$

I'm not sure how to tackle this problem. Consider the equivalence relation $R$ on $\Bbb N \times\Bbb N$ given by : $$(a, b)R(c, d) \iff a + d = b + c$$ (i) Show that $R$ is an equivalence ...
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### Book on foundational reasoning of standard arithmetics “curriculum”

I am interested in a book that is about arithmetics but the presentation is not just the known to all formulas but the foundational logic behind it. The closest example I can think about is the way ...
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### What is the denial of a statement in logic math?

I'm trying to get the hang of denials in logic in math. I would like to use these two examples: "Some people are honest and some people are not honest. (All people)" "No one loves everybody. (All ...
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### Can one use the Hilbert-Ackermann Consistency Theorem to prove the consistency of $PRA$?

In his textbook Mathematical Logic, Shoenfield states the Hilbert-Ackermann Consistency Theorem as follows: "Consistency Theorem (Hilbert-Ackermann): An open theory $T$ is inconsistent iff there is ...
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### Theorems not Formulable in Set Theory

Several sites I have been reading say that set theory is a good foundation for mathematics because virtually every theorem can be cast into a theorem in set theory. What is an example of a theorem ...
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### Continuum Hypothesis in formalized language. [closed]

The Continuum Hypothesis was advanced by Georg Cantor in 1878, before that Zermelo–Fraenkel set theory was stablished. "There is no set whose cardinality is strictly between that of the integers and ...
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### Creating mathematics vs building houses

I found the following quote in the book "Calculus" by Michael Spivak. (At the first page of Part 5,Epilogue, where he will discuss fields, construction of the real number, and uniqueness of the real ...
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### Proof that the minimal inductive set does not contain any more elements than $\{\emptyset, s(\emptyset), s(s(\emptyset)),s(s(s(\emptyset))),…\}$?

I am studying axiomatic (ZF) set theory and in particular the construction of the natural numbers from axiomatic set theory. Let me start by giving some introduction to my question. The axiom of ...
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### Prob. 7, Chap. 1 in Baby Rudin

Here's problem 7 in the exercises following Chap. 1 in Principles of Mathematical Analysis by Walter Rudin, 3rd edition: Fix $b > 1$, $y > 0$, and prove that there is a unique real number ...
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### Can the Recursion Theorem be proved in Peano Arithmetic?

A recursive function on $\mathbb{N}$ can be defined as follows: Given an element $a \in \mathbb{N}$ and a function $f:\mathbb{N}\rightarrow\mathbb{N}$, we can define a function ...
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### Prob. 6, Chapter 1 in Baby Rudin [duplicate]

Here's problem $6$ in Chapter $1$ in the book Principles of Mathematical Analysis by Walter Rudin, $3$rd edition: Fix a real number $b$, such that $b > 1$. $(a)$ If $m, n, p, q$ are integers, ...
### Step 6, Appendix to Chapter 1 in Baby Rudin: How to show that if $\alpha > 0^*$ and $\beta > 0^*$, then $\alpha \beta > 0^*$?
I'm reading Appendix to Chapter 1 in Principles of Mathematical Analysis by Walter Rudin, 3rd edition. In this appendix, Rudin gives a proof of Theorem 1.19 by constructing $\mathbb{R}$ from ...